Earticle

In this paper, we consider the delay partial differential equation of three dimensions with constant coefficients. We established the alternating direction difference scheme by the standard finite difference method, gave the order of convergence of the format and the expression of the difference scheme truncation errors.

The Agricultural Systems Application Platform (ASAP) provides bottom-up modelling and simulation environment for agricultural engineer. The purpose of this study is to expand usability of the ASAP to the second order partial differential equations: elliptic equations, parabolic equations, and hyperbolic equations. The ASAP is a general-purpose simulation tool which express natural phenomenon with capsulized independent components to simplify implementation and maintenance. To use the ASAP in continuous problems, it is necessary to solve partial differential equations. This study shows usage of the ASAP in elliptic problem, parabolic problem, and hyperbolic problem, and solves of static heat problem, heat transfer problem, and wave problem as examples. The example problems are solved with the ASAP and Finite Difference method (FDM) for verification. The ASAP shows identical results to FDM. These applications are useful to simulate the engineering problem including equilibrium, diffusion and wave problem.

Image segmentation based partial differential equation is very popular in image analysis and computer vision. This article presents a geometric active contour model based on local entropy for segmentation. In the proposed method, energy functional item of this model contains local energy and global energy items, and uses local entropy of each point on the contour line to determine the proportion of each kind of energy. The combined method allows the contour for the rapid evolution and convergence to the final edge after iterations. Experiment results on both synthetic and real images demonstrate the efficiency and accuracy of this model.

Image denoising is a basic problem in image processing, fourth order partial differential equation model is an image denoising method proposed for maintaining the balance of denoising and edge keeping. But when the image is severely contaminated, the denoising effect is not ideal, it may fuzzes the boundary at the same time, this paper puts forward an improved fourth order partial differential equation model, in order to overcome the contradiction between denoising and edge keeping, better denoising effect is obtained, and through the numerical simulation the results show that the method has good stability and practical value.

The different types of warranty policy have been established in order to fulfill the demand of manufacturers and the requirement of buyers so that a win-win situation could be acquired. However, when considering these warranty policies, an important concept to keep in mind is warranty reclaims. Warranty reclaims extend the scope of warranty activities beyond the walls of a single company to encompass suppliers, manufacturers, OEMs, distributors, dealers, repair centers, policy carriers, and customers. The present work is designed for a novel method to solve the nonlinear warranty reclaims in the form of diffusion equations. The main motivation for this work is that the warranty reclaims can be represented as the diffusion equations, and the diffusion equation is a partial differential equation which describes density dynamics in a material undergoing diffusion. The diffusion equation is also used to describe processes exhibiting diffusive-like behavior such as warranty reclaims. The approximate solution of this problem is calculated in the form of finite differences with easily computable terms. To represent the capability and reliability of the method, some automobile warranty cases have been illustrated.

Non-quantum state particle gave kinetic energy by the inertia interaction, while quantum state particle stores its potential energy produced by atomic (or molecular) bondings. Schrödinger equations are those for the non-quantum state particle. The equation for quantum state particles are derived for non-steady state and steady state. General relativity is completed by deriving the equations for quantum state particles. The two dimensional stress tensors in the partial differential equilibrium equations can be converted to one dimensional tensors per unit volume, which generate Laplacian. The Laplacian has exact analytical solution and needs boundary conditions. It gives us exact solution of Navier - Stokes’s equation.

In recent years, the theory of partial differential equations (PDE) for its rigorous mathematical theory foundation has been widely used in the fields of image processing. Medical imaging seeks to reveal internal structures hidden by the skin and bones, as well as to diagnose and treat disease. In order to detect the noise in the medical images, many models are studied, but the noises in medical images are much more complex than typical images. This paper introduces a new image noise detection approach using morphology and partial differential equations where are based on the morphology reconstruction with anisotropic diffusion to make full use of the advantage of Catte model. This proposed approach has been tested with the biomedical cell images with comparing with the Catte model, PM model and Canny model. The experimental results show that this proposed approach outperforms the other three models in terms of defined indicator and efficiency.

Fractional order has the characteristics of memory and non-locality and it is different with integer order. Therefore, fractional differential equations can be used to describe some abnormal natural phenomena. At the same time, how to solve the fractional order partial differential equation and differential equations with fractional order has become a very important research field. Besides analytic solution, it is also important to investigate the numerical methods for fractional differential equations. In the paper, fundamental solution of the time fractional partial differential equation has been deduced, which is derived by Furrier transform and Laplace transform. According to the simulation, there is little difference between numerical solution and the exact solution when the solution is the time variable function. The results show the validity of the method.

In this paper, an efficient method for image segmentation with the help of the fuzzy solutions of partial differential equations is presented. We first designed a Poisson equation model for image segmentation using fuzzy solution technology, which aims to find a fuzzy solution to satisfy precisely the PDEs. Then, an appropriate segmentation model was obtained to extract the boundary of objects, according to the numerical characteristic in fuzzy solving process. Comparison with the previous approaches is provides to validate the validity of the proposed method. Experimental results on synthetic and real-world images demonstrate that the proposed method has good performances in terms of speed, accuracy, robustness against Gaussian noise, and effectiveness in shadow.

In this paper, when N is a compact Riemannian manifold, we discuss the method of using warped products to construct Lorentzian metrics on $M=[a,\;b){\times}_f\;N$ with specific scalar curvatures.

We give a method to derive partial differential equations for the product of any two classical orthogonal polynomials in one variable and thus find several new differential equations. We also explain with an example that our method can be extended to a more general case such as product of two sets of orthogonal functions.

In this paper, when N is a compact Riemannian manifold, we discuss the method of using warped products to construct timelike or null future complete Lorentzian metrics on $M{\;}={\;}[\alpha,\infty){\times}_f{\;}N$ with specific scalar curvatures.

In this paper, when N is a compact Riemannian manifold, we discuss the method of using warped products to construct timelike or null future(or past) complete Lorentzian metrics on $M{\;}={\;}[a,{\;}{\infty}){\times}_f{\;}N$ with specific scalar curvatures.

The governing equations of atmospheric dispersion most often taking the form of a second-order partial differential equation (PDE). Currently, typical computational codes for predicting atmospheric dispersion use the Gaussian plume model that is an analytic solution. A Gaussian model is simple and enables rapid simulations, but it can be difficult to apply to situations with complex model parameters. Recently, a method of solving PDEs using artificial neural networks called physics-informed neural network (PINN) has been proposed. The PINN assumes the latent (hidden) solution of a PDE as an arbitrary neural network model and approximates the solution by optimizing the model. Unlike a Gaussian model, the PINN is intuitive in that it does not require special assumptions and uses the original equation without modifications. In this paper, we describe an approach to atmospheric dispersion modeling using the PINN and show its applicability through simple case studies. The results are compared with analytic and fundamental numerical methods to assess the accuracy and other features. The proposed PINN approximates the solution with reasonable accuracy. Considering that its procedure is divided into training and prediction steps, the PINN also offers the advantage of rapid simulations once the training is over.

In this paper, we propose a class of meshless collocation approaches for the solution of time dependent partial differential algebraic equations (PDAEs) in terms of a radial basis function interpolation numerical scheme. Kansa's method and the Hermite collocation method (HCM) for PDAEs are given. A sensitivity analysis of the solutions from different shape parameter c is obtained by numerical experiments. With use of the random collocation points, we have obtain the more accurate solution by the methods than those by the finite difference method for the PDAEs with index-2, i.e, we avoid the influence from an index jump of PDAEs in some degree. Several numerical experiments show that the methods are efficient.

The paper deals with the solution of some fractional partial differential equations obtained by substituting modified Riemann-Liouville derivatives for the customary derivatives. This derivative is introduced to avoid using the so-called Caputo fractional derivative which, at the extreme, says that, if you want to get the first derivative of a function you must before have at hand its second derivative. Firstly, one gives a brief background on the fractional Taylor series of nondifferentiable functions and its consequence on the derivative chain rule. Then one considers linear fractional partial differential equations with constant coefficients, and one shows how, in some instances, one can obtain their solutions on bypassing the use of Fourier transform and/or Laplace transform. Later one develops a Lagrange method via characteristics for some linear fractional differential equations with nonconstant coefficients, and involving fractional derivatives of only one order. The key is the fractional Taylor series of non differentiable function $f(x+h)=E_{\alpha}(h^{\alpha}{D_x^{\alpha})f(x)$.

In this paper, when N is a compact Riemannian manifold, we discuss the method of using warped products to construct timelike or null future (or past) complete Lorentzian metrics on $M=(-{\infty},{\;}\infty){\;}{\times}f^N$ with specific scalar curvatures.

We study a class of catalytic super Brownian motion X in 1-dimension. We show under some conditions of catalyst, the process X is absolutely continuous and we get a stochastic partial differential equation for X.

In the theory of the conditional Wiener integral, the integrand is a functional of the standard Wiener process. In this paper we consider a conditional function space integral for functionals of more general stochastic process and the generalized Kac-Feynman integral equation. We first show that the existence of a partial differential equation. We then show that the generalized Kac-Feynman integral equation is equivalent to the partial differential equation.

The existence of solutions for nonlinear elliptic partial differential equations with general flux and damping terms is investigated.